Music as Science
Isaac Newton and the Genius of Johann Sebastian Bach
Christoph Wolff

What constitutes the artistic legacy of the Leipzig Thomascantor, Johann Sebastian Bach? To answer, let me quote the passionate pronouncement made by Bach's son, Carl Philipp Emanuel, and his former student, Johann Friedrich Agricola, in their obituary of "The World-Famous Organist, Mr. Johann Sebastian Bach" (written 1750, published 1754):

If ever a composer showed polyphony in its greatest strength, it was certainly our late lamented Bach. If ever a musician employed the most hidden secrets of harmony with the most skilled artistry, it was certainly our Bach. No one ever showed so many ingenious and unusual ideas as he in elaborate pieces such as ordinarily seem dry exercises in craftsmanship.

More often than not, superlatives such as these provoke skepticism, but this obituary statement—though penned under the immediate burden of loss and pressure of time—presents a remarkably insightful summary of Bach's most important musical accomplishments. It emphasizes that his music truly demonstrates the power of polyphony, the artful application of intrinsic harmonic structure and organization, and his imaginative, uncommon, and original approach in the design of complex works.

The obituary statement associating Bach's music with "polyphony in its greatest strength," the employment of "the most hidden secrets of harmony with the most skilled artistry," and "ingenious and unusual ideas" pervading "elaborate pieces" has an extremely positive ring, yet it must be understood against the well-known background of criticism to which Bach was subjected—most directly and vocally in an infamous attack by a certain Johann Adolph Scheibe in the year 1737:

This great man would be the admiration of whole nations if he had more amenity, if he did not take away the natural element in his pieces by giving them a turgid and confused style, and if he did not darken their beauty by an excess of art.

Johann Abraham Birnbaum, a Leipzig friend of Bach and his "ghostwriter," explicitly articulates the composer's views on the matter of art and nature, elegantly countering Scheibe's broadside:

This sentence contradicts the nature of true art, which is what is here being discussed. The essential aims of true art are to imitate nature, and, where necessary, to aid it. [...] Thus art lends nature a beauty it lacks, and increases the beauty it possesses. Now, the greater the art is—that is, the more industriously and painstakingly it works at the improvement of nature—the more brilliantly shines the beauty thus brought into being. Accordingly it is impossible that the greatest art should darken the beauty of a thing.

Birnbaum's argument against Scheibe emphasizes the ancient Aristotelian principle "art imitates nature" (ars imitatur naturam), a dictum that lay at the heart of what Bach, like most of his contemporaries, considered musical science. This term "musical science" (musicalische Wissenschaft) occurs most prominently in Bach's letter addressed to the Elector of Saxony and King of Poland, Friedrich August II, in conjunction with the dedication in 1733 of the Kyrie and Gloria of what later became the B-minor Mass. Here the composer writes: "To your Royal Highness I submit in deepest devotion the present small work of science which I have achieved in musique." A Mass as a work of musical science? Indeed.

Like no other work of Bach’s, the Kyrie-Gloria Mass of 1733 and, to an even greater extent, the completed B-minor Mass, represents a summary of his writing for voice, not only in its variety of styles, compositional devices, and range of sonorities, but also in its high level of technical polish. The Mass offers a full panoply of the art of musical composition, a comprehensive grasp of music history, particularly in its use of old and new styles. Just as theological doctrine survived over the centuries in the words of the Mass, so Bach's mighty setting preserved the musical and artistic creed of its creator for posterity.

What was the intellectual basis of Bach’s musical and artistic creed? There is no question that Bach was influenced by the climate of intellectual inquiry and search for truth propounded by Gottfried Wilhelm Leibniz and Christian Wolff. They defined philosophy—according to Bach's student Lorenz Christoph Mizler—as "a science of all things that teaches us how and why they are or can be." Especially in Leipzig, Bach was exposed to much abstract theoretical discourse, but he had no interest in contributing to it himself. He was no theoretician. Instead, he focused on practice, on a genuinely empirical approach in which he explored "the most hidden secrets of harmony with the most skilled artistry." That is, Bach's interest was to push and expand the known limits of musical composition.

The intricate musical art resulting from this approach figured in a public literary dispute between the Berlin capellmeister Agricola, co-author of the obituary referred to earlier, and Filippo Finazzi, an opera singer in Hamburg, that recalls the earlier Scheibe-Birnbaum controversy. In August 1750, just days after Bach's death, Agricola wrote that Finazzi denied Bach's music:

the effect of pleasure for the listener who would not savor such difficult harmony. Yet, assuming the harmonies [i.e., compositions] of this great man were so complex that they would not always achieve the intended result, they nevertheless serve for the connoisseur's genuine delight. Not all learned people are able to understand a Newton, but those who have progressed far enough in profound science so they can understand him will find the greater gratification and real benefit in reading his work.

Here, for the first time, a parallel is drawn between Bach and Isaac Newton—not by way of analogies between Bach's music and Newton's physical science, but by explaining that Bach's music is best appreciated by real connoisseurs, just as Newton's writings are best understood by readers with a profound knowledge of science. Newton, a generation older than Bach, had earned a legendary reputation across Europe by the early eighteenth century, and by 1750 he represented the undisputed paradigm of the scien­tist as genius. In 1740 Lorenz Christoph Mizler called him "the immortal Newton." The British mathematician, physicist, and philosopher had died only in 1727, but ever since 1714, when the Leipzig periodical, Acta eruditorum, published one of the most important early reviews of his principal opus, with a careful collation of the 1687 and 1713 editions of the Philosophiae Naturalis Principia Mathematica, Leipzig University had been the center of Newtonianism in Germany. One of its most prominent representatives was Johann Heinrich Winckler, a colleague of Bach's at the Thomasschule, and the author of a Bach cantata text who was famous for his experiments in electricity and among the first German scientists elected to the Royal Academy in London. The world of "the immortal Newton" was Bach's world.

Newton's theoretical and experimental works exemplified a new kind of scientific method, though without differentiating clearly between the roles played in this endeavor by reason and by observation. Moreover, Newton's understanding that his discoveries "pointed to the operations of God" is typical of his pre-Enlightenment outlook. Unlike modern—post-Enlightenment—science which focuses only on the knowledge of nature, in Newton's view the search for truth always encompasses both natural and divine principles. Trying to understand the relationship between God and nature led him to explore the boundaries between them, where he ultimately saw the fusion of natural and divine principles.

Newton created a revolution in a number of areas, most notably the calculus and the theory of mechanics. He was recognized for it by his contemporaries, and altogether his work represented the pinnacle of the seventeenth-century Scientific Revolution. Bach, on the other hand, created no revolution, but then, the stakes were completely different. In the search for scientific truth, the principle of universal gravity, for example, would have been discovered eventually by someone, even if that someone had not been Newton. However, the search for "artistic truth"—so to speak—in the exploration of both the natural fundaments and the limits of musical composition is not guided by the classical two-valued logic leading to a true or false result. The element of individuality plays too decisive a role in all artistic endeavors, both pre- and post-Enlightenment. Yet Bach's music, his life work, in part subconsciously and in part consciously, was affected more than that of any other contemporary musician by the spreading culture of Newtonianism and the general spirit of discovery that prevailed following the Scientific Revolution, a spirit that no bright and keen intellect could escape. And under the firm umbrella of seventeenth-century Lutheran theology, Bach's musical discoveries—like Newton's, whose works Bach almost certainly did not know—ultimately pointed to the operations of God.

The eclectic natural philosophy of late seventeenth- and early eighteenth-century Germany defined itself as "a science of all things that teaches us how and why they are or can be." Bach's musical philosophy might well be understood analogously—not in terms of subject matter and methodology, but in the sense of an empirical search for the foundation and boundaries of his art—as the science of all musical phenomena that teaches us not only how and why they are or can be, but also how they relate to an understanding of nature, God's creation, that is, Newton's "system of the world." Moreover, the sheer scope and breadth of Newton's intellectual endeavors find their analogy in the enormous and unparalleled range of the interests, undertakings, and efforts of Bach the musician.

To this point, I have discussed all these matters more or less in abstract terms. Let me now discuss five specific examples to illuminate some of the points made previously. I will focus on examples that demonstrate how Bach moves within self-imposed, extremely narrow constraints, challenging himself to explore, research, and uncover "the secrets of harmony" even in the most fundamental materials given by nature, basic materials such as triad and scale. Thinking of Birnbaum's notion of "art that lends nature the beauty it lacks," Bach's examples demonstrate that it is indeed possible "to improve nature."

[In the lecture, the following examples were presented as recorded excerpts from the compositions referred to below. Comments were amplified by illustrations at the piano.]

EXAMPLE 1: Canon 'Trias harmonica," BWV1072

I begin with Bach's shortest composition, a piece that is hardly ever discussed and certainly never performed: an enigmatic canon entitled "Trias harmonica" (the harmonic triad), published in Friedrich Wilhelm Marpurg's Abhandlungvon derFuge (Berlin, 1753). The abbreviated, enigmatic notation specifies what the structure is supposed to be, namely, "a canon for eight voices and two choirs" in plain C major. The composer demonstrates with this composition what constitutes a harmonic triad: not a vertical C-major chord, but a structure resulting from accumulated counterpoint, however basic the melody is (in this case, the scale pattern c-d-e-f-g). In many ways the piece represents Bach's musical Credo.

What you hear is, of course, musical theory, not real music; but it tells you a lot about Bach's mind and about the internal control factors that guide his genuinely musical composition. In this connection, let me turn to Birnbaum who in 1738 wrote defending Bach against Scheibe: "It is certain [...] that the voices in the works of this great master of music work wonderfully in and about one another, but without the slightest confusion. They move along together in opposition, as necessary. They part company, and yet all meet again at the proper time. Each voice distinguishes itself clearly from the others [... ] They now flee, now follow one another without one's noticing the slightest irregularity in their efforts [...]."

EXAMPLE 2: Fantasia in c minor, BWV 906

My second example introduces a different configuration of basic materials, put very effectively together in a keyboard fantasy. The constituent elements are: (1) a C-minor triad—plain and arpeggiated, ascending and descending; (2) two alternating scale-patterns, diatonic and chromatic, ascending and descending. We notice here a further dimension, that of keyboard virtuosity. Bach finds himself in his very domain, playing with the most fashionable manner of crossing hands. This piece croise combines fundamental materials, triad and scale-based, with fanciful virtuosity. At the same time, the work does not display empty virtuosity at all, for it is very tightly constructed and demonstrates what can be done by uncovering the true potential even of these rudimentary elements.

EXAMPLE 3: Cantata "Nach dir, Herr, verlanget mich," BWV 150/4

The next example introduces a vocal piece, the earliest extant cantata Bach composed in Arnstadt, ca. 1705. The fourth (and shortest) section of the multi-sectional work is a setting of a single Biblical line, a verse from Psalm 25: "Leite mich nach deiner Wahrheit" (Lead me in thy truth). The two key words, the verb "lead" and the noun "truth," have the young Bach turn to a musical idea he himself did not invent, an element he considered given by nature, by God: the scale. But the way he treats the scale as the centerpiece, the musical axis of the setting, is quite extraordinary. The axis moves straight across the score and, thereby, penetrates and affects all voices: bass, tenor, alto, soprano; violins move through three full octaves; all voices share in the scale as their firm structural backbone. The musical and theological message: Truth and divine guidance are inseparable from one another.

EXAMPLE 4: Fantasia in g minor, BWV 542/1

Example 4 introduces yet another solution of dealing with fundamental material in a highly innovative manner. In this organ fantasia, Bach researches the possibilities of harmonizing the descending scale, approaching with uncompromising logic and resolution a pivot point: a diminished chord, a dramatic fermata. After that, things turn around, and instead of a simple resolution the harmonic structure gets involved with extreme chromaticism. This work shows the composer of the Well-Tempered Clavier at work, a composer who breaks through the limits of the conventional tonal system. Moreover, it is amazing to realize how Bach creates the illusion of an endless descending scale by breaking the pedal octaves in a camouflaged mode.

EXAMPLE 5: "Et incarnatus est," from the B-minor Mass, BWV 232

Finally, an example that in all likelihood represents one of Bach's last compositions, if not the very last: the "Et incarnatus est" from the B-minor Mass. The movement represents an afterthought that occurred to the composer while reviewing and revising his large-scale Mass in 1749, about a year before his death. The text of this section was originally incorporated in the "Et in unum" duet that was followed directly by the "Crucifixus" movement. However, Bach apparently thought that the text dealing with the mystery of the incarnation ("and was incarnate by the Holy Ghost of the Virgin Mary, and was made man") deserved special and separate treatment as this portion of the Mass text historically always drew special musical attention. Therefore, he re-texted the duet and inserted into the score an extra leaf with the newly composed "Et incarnatus est."

The basic musical idea of the movement is again of a rather simple design: a B-minor triad extends over a pedal point, reinforced by a steady instrumental obbligato (unison violins) that accentuates the triadic tones with emphatic appoggiaturas. This straight-forward structural outline enables the five-part choral setting to enter in crystal clear declamation. The point of imitation is based on descending triadic formation that is modified with every voice entry and finally, with the bass entry, forces the pedalpoint to change from B to B-sharp. A rational musical setting of utterly irrational non-musical content results in a movement with a thoughtfully interpretive and highly expressive theological message.

We recognize in these five examples Bach's remarkable ability to integrate and synthesize the various parameters and components of his musical science and his highly developed sense for the creation of unified structures. Although he realized the significance of theoretical discourse and apparently encouraged his students to engage therein, he himself made no contribution to it, and focused instead on "practical elaborations" for the instruction and delight of "those who have a concept of what is possible in art and who desire original thought and its special, unusual elaboration"—as Carl Philipp Emanuel Bach put it in advertising the first edition of The Art of Fugue in the year following his father's death.

The limited knowledge of Bach’s works that were available after 1750—The Art of Fugue and The B-minor Mass figured prominently among them—only magnified the effect of individual compositions as their musical essence and compositional makeup were contemplated. Bach's music immediately established new benchmarks of compositional artistry and technical perfection. The exemplary value of Bach's music was recognized as each work soon became a touchstone for performers, composers, and theorists alike—a distinction the pieces hold to this day. After Bach, music was no longer the same. A paradigm shift had taken place and gradually took hold, comparable to what happened in science as a result of Newton's work. The two men reached pinnacles of a very different kind, but they lived, thought, and worked in the same intellectual climate of scientific discovery and empirical testing of fundamental principles.

Not just among composers of his time, but among composers in general, Bach was one of the most active, dedicated, and prolific teachers the world has seen. Many of his students disseminated his music and teachings, quite a number of them having become influential authors of theoretical treatises. In fact, soon after 1750, German music theory—and half a century later European theory as well—reoriented itself almost solely because of the prevailing influence of "the Bach School." If Bach ever created a "revolution" it was in his teaching of composition by fully integrating the principles of thorough bass, harmony, and counterpoint, elements that had previously been treated separately. Major models for this method were established primarily by two works: first, the Well-Tempered Clavier that defined the princi­ples of free and strict composition (manuscript copies circulated widely until 1801 when three editions appeared independently in Leipzig, Bonn, and Zurich), and second by the collection of 370-plus four-part chorales that charted the course for tonal harmony (two posthumous editions were published, one in the late 1760s and the other in the mid-1780s).

The impact of Bach's music and teachings was such that, even at a time when the focus of the music world had clearly shifted to the Vienna of Haydn and Mozart, the musician and critic Christian Friedrich Daniel Schubart wrote on Bach's significance, referring to him—in an oblique yet obvious juxtaposition to Handel, the "Orpheus Brittannicus"—as "the Orpheus of the Germans!" In the same essay, he calls Bach "a genius of the highest order" and puts him on a par with the paradigmatic scientist-genius when he states, "What Newton was as a philosopher, Bach was as a musician." In other words, Newton brought about fundamental changes and established new principles in the world of science, and Bach did the same in the world of music, both in composition and in performance.

At the turn of the nineteenth century, Schubart's appraisal was not only echoed in the Allgemeine musikalische Zeitung, the leading music periodical of the day, to which Beethoven and others subscribed, but was further concretized in terms of Bach as "lawmaker":

The name of Johann Sebastian Bach radiates supremely and sublimely above those of all German composers in the first half of the past century. He embraced with Newton's spirit everything that has hitherto been thought about harmony [composition] and that has been presented as examples thereof, and he penetrated its depths so completely and felicitously that he must be justly regarded as the lawmaker of genuine harmony, which is valid up to the present day.

Indeed, the "progenitor of harmony," as Beethoven put it, has not lost his attraction today. Even though we cannot and must not see all of Bach's music under the perspectives of his eighteenth-century interpreters and admirers, our understanding of his musical philosophy benefits considerably from placing it in the intellectual milieu of the Newtonian spirit of discovery. But different from scientific discoveries, Bach's art of penetrating, exhausting, expanding, and transcending all conceivable possibilities of harmony, that is, of musical composition, is by no means to be understood as theoretical exercise; it is a spirit of musical discovery that reaches beyond pure intellect by speaking directly to the heart. May the spirit of discovery embedded in Bach's art also guide our contemporary musical practices.


Christoph Wolff of Harvard University, a Distinguished Advisor to the Bach Institute, presented this essay as a lecture during the inaugural festivities of the Institute. Wolff is the author of the widely acclaimed Johann Sebastian Bach: The Learned Musician, discussed later in this issue.

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